SYMPLECTIC (−2)-SPHERES AND THE SYMPLECTOMORPHISM GROUP OF SMALL RATIONAL 4-MANIFOLDS II

We study the symplectic mapping class groups of (CP²#5CP², ω). Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures J₀ as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by Wu [Math. Ann. 359 (2014), pp. 153–168]) and the study of a semi-toric model to understand a “connecting map”, whose cokernel is the symplectic mapping class group. Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms ω. Let N_ω be the number of (−2)-symplectic spherical homology classes. Torelli SMCG is trivial if N_ω > 8; it is π0(Diff⁺(S², 5)) if N_ω = 0 (by Seidel [Lecture notes in Math., Springer, Berlin, 2008] and Evans [J. Symplectic Geom. 9 (2011), pp. 45–82]); and it is π0(Diff⁺(S², 4)) in the remaining case. Further, we completely determine the rank of π1(Symp(CP²#5CP², ω)) for any given symplectic form. Our results can be uniformly presented in terms of Dynkin diagrams of type A and type D Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds (open problem 16 in McDuff-Salamon’s book 3rd version [Oxford graduate texts in mathematics, Oxford University Press, Oxford, 2017]).